The Rule of 72

Logarithms, Doubling, the Rule of 72, and Finance

One of the most important constants in finance is the natural logarithm of 2, which is approximately 0.693147.

Why does this matter? Because doubling lies at the heart of investing, compound growth, interest rates, inflation, and risk analysis. The mathematics behind doubling leads naturally to logarithms, and from there to the famous Rule of 72.

A Simple Infinite Series for log(1 + r)

A basic infinite series for log(1 + r) is:

log(1 + r) = r - r^2/2 + r^3/3 - r^4/4 + r^5/5 - ...

This series works when r is between -1 and 1, with convergence at r = 1 but divergence at r = -1.

For small values of r, the higher powers become very small, so we get the approximation:

log(1 + r) is approximately r

This simple fact is one reason logarithms are so useful in finance.

Solving a = log(1 + r) for r

Suppose you know that:

a = log(1 + r)

Then exponentiating both sides gives:

e^a = 1 + r

So:

r = e^a - 1

This lets us move back and forth between log returns and ordinary percentage returns.

Simple Returns Versus Log Returns

If an asset price changes from P0 to P1, then the simple return is:

r = (P1 - P0) / P0

The log return is:

a = log(P1 / P0)

Since P1 / P0 = 1 + r, we get:

a = log(1 + r)

Conversely:

r = e^a - 1

Log returns are especially useful because they add neatly over time. If a price moves from P0 to P1 and then from P1 to P2, the total log return is just the sum of the two smaller log returns.

Why log(1 + r) Is Close to r for Small Returns

Using the series:

log(1 + r) = r - r^2/2 + r^3/3 - ...

For small r, this becomes approximately:

log(1 + r) is approximately r

A more accurate approximation is:

log(1 + r) is approximately r - r^2/2

This correction helps explain volatility drag in investing. A volatile investment may have an attractive average return, but its long-run compounded growth can still be lower than expected.

Continuous Compounding

If an investment grows continuously at annual log rate a, then its value after time t is:

P(t) = P0 times e^(a t)

The equivalent ordinary annual return is:

r = e^a - 1

For example, if a = 0.05, then:

r = e^0.05 - 1, which is about 0.05127

So a continuously compounded rate of 5 percent corresponds to an ordinary annual return of about 5.127 percent.

Doubling Time and log(2)

Suppose an investment doubles after T years. Then:

(1 + r)^T = 2

Taking natural logarithms gives:

T times log(1 + r) = log(2)

So the exact doubling time is:

T = log(2) / log(1 + r)

This is where log(2) enters finance in a fundamental way. Every doubling calculation ultimately contains this constant.

Example: What Annual Rate Doubles Money in 8 Years?

If an investment doubles in 8 years, then:

(1 + r)^8 = 2

So:

1 + r = 2^(1/8)

and therefore:

r = 2^(1/8) - 1

Numerically, this is about:

r = 0.0905

So the annual rate is about 9.05 percent.

The equivalent continuously compounded annual rate is:

a = log(2) / 8

which is about 0.08664, or 8.664 percent.

Deriving the Rule of 72

The Rule of 72 says:

Years to double is approximately 72 divided by the interest rate in percent

Equivalently:

Interest rate in percent is approximately 72 divided by the number of years to double

Here is where it comes from.

Start with the exact formula:

T = log(2) / log(1 + r)

For modest rates of return, log(1 + r) is approximately r, so:

T is approximately log(2) / r

Since log(2) is about 0.693, this becomes:

T is approximately 0.693 / r

If r is written as a percentage R, so that r = R / 100, then:

T is approximately 69.3 / R

So the mathematically direct rule would be the Rule of 69.3.

Why 72 Instead of 69.3?

There are two main reasons.

First, 72 is easier to divide mentally than 69.3. It works well with many common interest rates such as 2, 3, 4, 6, 8, 9, and 12.

Second, using 72 partly compensates for the fact that log(1 + r) is not exactly equal to r. The next term in the series is negative:

log(1 + r) is approximately r - r^2/2

This makes the true doubling time a bit longer than the simplest estimate suggests, so bumping the constant from 69.3 up to 72 often improves the mental estimate in the interest-rate ranges people commonly use.

Examples

At 6 percent, the Rule of 72 gives:

72 / 6 = 12 years

The exact doubling time is about 11.9 years.

At 8 percent, the Rule of 72 gives:

72 / 8 = 9 years

The exact doubling time is about 9.0 years.

At 9 percent, the Rule of 72 gives:

72 / 9 = 8 years

The exact doubling time is about 8.04 years.

At 12 percent, the Rule of 72 gives:

72 / 12 = 6 years

The exact doubling time is about 6.12 years.

So the Rule of 72 is not exact, but it is often impressively useful.

Applications in Finance

1. Estimating investment growth

Investors use the Rule of 72 to estimate how long it will take a portfolio, savings account, bond ladder, business revenue stream, or dividend stream to double.

2. Inflation

The same rule applies to inflation. If prices rise at 6 percent per year, then the overall price level will roughly double in about 12 years. That means purchasing power is roughly cut in half over that period.

3. Comparing ordinary and continuous compounding

The equation a = log(1 + r) lets analysts move back and forth between ordinary annual returns and continuously compounded returns. This is useful in fixed income, derivatives, and mathematical finance.

4. Quantitative finance and log returns

Many quantitative models use log returns because they add over time and behave more cleanly in probabilistic models than simple returns do.

5. Volatility drag

Because log(1 + r) is approximately r - r^2/2, volatility can reduce long-run geometric growth even when average arithmetic returns look appealing. This idea matters in portfolio theory, leverage analysis, and long-term investment strategy.

Why log(2) Matters So Much

The constant log(2) appears whenever doubling is involved. That includes compound interest, inflation, population growth, and information theory.

In finance, its meaning is especially concrete: it is the logarithmic size of one doubling.

Conclusion

The simple series

log(1 + r) = r - r^2/2 + r^3/3 - ...

opens the door to a great deal of practical finance. It explains why log returns and ordinary returns are close for small moves, why continuously compounded rates are convenient, why doubling time is governed by log(2), and why the Rule of 72 works as well as it does.

So when someone says an investment will double in eight years, there is elegant mathematics underneath the claim:

(1 + r)^T = 2

T = log(2) / log(1 + r)

r is approximately 72 / T percent

And behind all of it sits one of the most useful constants in applied mathematics and finance:

log(2) is approximately 0.693147